Residue and signature group of a modulus ($G_{\mathfrak{c}}=({\mathcal{O}}_K/{\mathfrak{c}}_0{)}^{\times }\times \{\pm 1{\}}^{{\mathfrak{c}}_{\infty }}$)

This group records finite unit residues and signs at the selected real places of a modulus of a number field. Omit the residue factor for trivial finite modulus. Its quotient by the image of the unit group is naturally the kernel of the map from the ray class group to the ordinary ideal class group. The weak approximation for number fields theorem supplies elements realizing the specified data, and units account for changing a generator of a principal fractional ideal.